The Sturm-Liouville problem and the Polar Representation Theorem

Details The Sturm-Liouville problem and the Polar Representation Theorem The Sturm-Liouville problem and the Polar Representation Theorem

2010-06-29

arxiv.org/abs/1006.5718v1

International Journal of Pure and Applied Mathematics, Vol. 61, No. 4, 2010, 467-494

Mathematics

Classical Analysis and ODEs

Scientific paper

The polar representation theorem for the n-dimensional time-dependent linear Hamiltonian system with continuous coefficients, states that, given two isotropic solutions (Q1, P1) and (Q2, P2), with the identity matrix as Wronskian,the formula Q2 = rcos(f), Q1 = rsin(f), holds, where r and f are continuous matrices, r is non-singular and f is symmetric. In this article we use the monotonicity properties of the matrix f eigenvalues in order to obtain results on the Sturm-Liouville problem.

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